Consider the coupled system of differential equations: frac{dx_1}{dt} = 2x_1 - x_2 frac{dx_2}{dt} = 4x_1 + 2x_2 (a) Let A = egin{pmatrix} 2 & -1 \ 4 & 2 end{pmatrix} Determine the eigenvalues and eigenvectors. (15 marks) (b) Find the unique solution of the coupled system of differential equations that satisfies the initial conditions x_1(0) = 1, x_2(0) = -1. (10 marks)
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Now, let's find the eigenvectors corresponding to each eigenvalue. For $\lambda_1 = 6$: $$ (A - 6I)v_1 = 0 $$ $$ \begin{pmatrix} -4 & 4 \\ 4 & -4 \end{pmatrix} \begin{pmatrix} v_{11} \\ v_{12} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} $$ We can see Show more…
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Consider the system {x'1 = 4x1 + x2', x'2 = 6x1 - x2}. a) Express the system in matrix form: x' = Ax. b) Solve the characteristic equation (A - λI) = 0 to find the two real eigenvalues λ1 and λ2. c) Find the eigenvectors v1 and v2 corresponding to the eigenvalues λ1 and λ2 by solving [A - λI]v = 0. d) Express the two linearly independent solutions x1 and x2 in the form xi = e^(λit)vi. e) Express the general solution as a linear combination of x1 and x2.
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